Question

If the focal length of a convex lens is 1 m, what is the maximum thickness of the lens?

Answer 1

Answer on Question #59194, Physics / Optics

If the focal length of a convex lens is $1\,\mathrm{m}$, what is the maximum thickness of the lens?

Find: $d_{\mathrm{max}}$?

Given:

$f = 1\,\mathrm{m}$

Solution:

For not thin lenses fair value:

$$\frac{n_{\mathrm{environment}}}{f} = \left(n_{\mathrm{lens}} - n_{\mathrm{environment}}\right) \times \left(\frac{1}{R_1} - \frac{1}{R_2} + \frac{(n_{\mathrm{lens}} - n_{\mathrm{environment}})\mathrm{d}}{n_{\mathrm{lens}} R_1 R_2}\right) \quad (1),$$

where $n_{\mathrm{lens}}$ – absolute index of lens' refractive,

$n_{\mathrm{environment}}$ – absolute index environment' refractive,

$f$ – focal length,

$R_1, R_2$ – radii of curvature of the lenses surfaces

By task: $R_1 > 0, R_2 > 0$ (2)

Believe that lens is symmetric: $R_1 = R_2 = R$ (3)

(2) and (3) in (1): $\frac{n_{\mathrm{environment}}}{f} = \left(n_{\mathrm{lens}} - n_{\mathrm{environment}}\right) \times \left(\frac{(n_{\mathrm{lens}} - n_{\mathrm{environment}})\mathrm{d}}{n_{\mathrm{lens}} R^2}\right)$ (4)

Of (4) $\Rightarrow d = \frac{n_{\mathrm{lens}} R^2 n_{\mathrm{environment}}}{f(n_{\mathrm{lens}} - n_{\mathrm{environment}})^2}$ (5)

We believe that a lens is glass and placed in the air.

$n_{\mathrm{air}} = 1,0$ (6)

Tabular data: $n_{\mathrm{glass}} = 1,5 - 1,9$ (7)

Of (5) and (7) $\Rightarrow d_{\mathrm{max}}$ if $n_{\mathrm{glass}}$ minimum

Of (5) $\Rightarrow$ if $n_{\mathrm{glass}} = 1,5$ than $d_{\mathrm{max}}$

Of (5) $\Rightarrow d_{\mathrm{max}} = \frac{6R^2}{f}$

Answer:

The general formula: $d = \frac{n_{\mathrm{lens}} R^2 n_{\mathrm{environment}}}{f(n_{\mathrm{lens}} - n_{\mathrm{environment}})^2}$

The simplified formula ($n_{\mathrm{air}} = 1,0$): $d = \frac{n_{\mathrm{lens}} R^2}{f(n_{\mathrm{lens}} - 1)^2}$

Maximum thickness ($n_{\mathrm{glass}} = 1,5$): $d_{\mathrm{max}} = \frac{6R^2}{f}$

Numeric value of maximum thickness for this task ($f = 1\,\mathrm{m}$): $\{d_{\mathrm{max}}\} = 6R^2$

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